Problem 111 - Precalculus with Limits

Question

In Exercises 103-112, use sigma notation to write the sum. \( \dfrac{1}{4} + \dfrac{3}{8} + \dfrac{7}{16} + \dfrac{15}{32} + \dfrac{31}{64} \)

Step-by-Step Solution

Answer

The sigma notation for the given series is \( \sum_{i=0}^{4} \frac{1}{4*2^i} \).

Step 1: Identification

Identify the first term \(a_1\), the common ratio \(r\), and the number of terms \(n\). Here, \(a_1 = 1/4\), \(r = 1/2\) and the number of terms n=5.

Step 2: Constructing Sigma Notation

According to the properties of geometric series, it can be written in the sigma notation as: \(\sum_{i=0}^{n-1} a_1 * r^i \). Substituting the identified values into this formula, we get: \(\sum_{i=0}^{4} \frac{1}{4} * (\frac{1}{2})^i \).

Step 3: Finalize the Sigma Notation

Simplify the summation expression to obtain the final answer: \( \sum_{i=0}^{4} \frac{1}{4*2^i} \). This expression is equal to the given series, written in sigma notation.

Key Concepts

Understanding Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The structure looks like this: first term, followed by the first term multiplied by the common ratio, then the next term is that result multiplied by the common ratio again, and so on. This defines a growing or shrinking pattern based on the ratio:

• If the common ratio is greater than 1, each term will be larger than the last.
• If the common ratio is between 0 and 1, each term will be smaller than the last.
• If the common ratio is negative, the terms will alternate in sign.

Understanding geometric series helps in a wide range of mathematical applications, such as investment calculations, population models, and computer algorithms. To write the series in a compact form, sigma notation can be employed, which efficiently represents the sum of the series without listing out all the terms.

Identifying the First Term of a Series

The first term of a series, often denoted as \(a_1\), is crucial because it serves as the foundation for constructing the entire sequence. In a geometric series, this first term establishes the starting point from which subsequent terms are generated through repeated multiplication by the common ratio.
In the example provided, the first term \(a_1\) is \(\frac{1}{4}\). This means that the series begins with this fraction, and each subsequent term is built by multiplying it by the common ratio (which we'll explore more in the following section). Determining \(a_1\) accurately is essential, as it directly influences the values of every other term in the sequence.
Whether you have a handful of terms or a larger set, the first term remains a necessary component for identifying the precise nature of the sequence.

Exploring the Common Ratio of a Series

The common ratio of a series, represented by \(r\), is a multiplier that explains how each term in the series is related to the previous one. In a geometric series, this is a constant value that remains unchanged throughout the sequence. Finding this ratio is done by dividing any term in the series by the preceding term.
In the given example, we identified the common ratio \(r\) as \(\frac{1}{2}\). This means each term in the series is half the previous term, showcasing a shrinking pattern. Knowing the common ratio allows us to predict the entire series progression:

• Given the starting term and the common ratio, the whole series can be efficiently constructed using the formula \(a_1 * r^i\), where \(i\) represents the term number.
• Being able to identify and apply the correct common ratio is key to manipulating and understanding geometric sequences.

Mastering the concept of the common ratio enables ease in handling complex series-related problems and recognizing patterns inherent in various mathematical contexts.